Theorem pmtrsn 19485

Description: The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)

Assertion

Ref	Expression
pmtrsn	⊢ (pmTrsp‘{𝐴}) = ∅

Proof of Theorem pmtrsn

Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5368	. . 3 ⊢ {𝐴} ∈ V
2		eqid 2739	. . . 4 ⊢ (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
3	2	pmtrfval 19416	. . 3 ⊢ ({𝐴} ∈ V → (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
4	1, 3	ax-mp 5	. 2 ⊢ (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
5		eqid 2739	. . . . 5 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6	5	dmmpt 6191	. . . 4 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
7		2on0 8409	. . . . . . . . 9 ⊢ 2o ≠ ∅
8		ensymb 8939	. . . . . . . . . 10 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅)
9		en0 8955	. . . . . . . . . 10 ⊢ (2o ≈ ∅ ↔ 2o = ∅)
10	8, 9	bitri 276	. . . . . . . . 9 ⊢ (∅ ≈ 2o ↔ 2o = ∅)
11	7, 10	nemtbir 3030	. . . . . . . 8 ⊢ ¬ ∅ ≈ 2o
12		snnen2o 9145	. . . . . . . 8 ⊢ ¬ {𝐴} ≈ 2o
13		0ex 5229	. . . . . . . . 9 ⊢ ∅ ∈ V
14		breq1 5075	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ≈ 2o ↔ ∅ ≈ 2o))
15	14	notbid 319	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (¬ 𝑦 ≈ 2o ↔ ¬ ∅ ≈ 2o))
16		breq1 5075	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝑦 ≈ 2o ↔ {𝐴} ≈ 2o))
17	16	notbid 319	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2o ↔ ¬ {𝐴} ≈ 2o))
18	13, 1, 15, 17	ralpr 4632	. . . . . . . 8 ⊢ (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o ↔ (¬ ∅ ≈ 2o ∧ ¬ {𝐴} ≈ 2o))
19	11, 12, 18	mpbir2an 717	. . . . . . 7 ⊢ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o
20		pwsn 4831	. . . . . . . 8 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
21	20	raleqi 3295	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o)
22	19, 21	mpbir 232	. . . . . 6 ⊢ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o
23		rabeq0 4316	. . . . . 6 ⊢ ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o)
24	22, 23	mpbir 232	. . . . 5 ⊢ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅
25	24	rabeqi 3404	. . . 4 ⊢ {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
26		rab0 4314	. . . 4 ⊢ {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
27	6, 25, 26	3eqtri 2766	. . 3 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
28		mptrel 5768	. . . 4 ⊢ Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
29		reldm0 5870	. . . 4 ⊢ (Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅))
30	28, 29	ax-mp 5	. . 3 ⊢ ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅)
31	27, 30	mpbir 232	. 2 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
32	4, 31	eqtri 2762	1 ⊢ (pmTrsp‘{𝐴}) = ∅

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391  Vcvv 3431   ∖ cdif 3880  ∅c0 4261  ifcif 4454  𝒫 cpw 4529  {csn 4555  {cpr 4557  ∪ cuni 4838   class class class wbr 5072   ↦ cmpt 5153  dom cdm 5618  Rel wrel 5623  ‘cfv 6485  2_oc2o 8389   ≈ cen 8880  pmTrspcpmtr 19407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-1o 8395  df-2o 8396  df-er 8633  df-en 8884  df-pmtr 19408

This theorem is referenced by:  psgnsn  19486